www.nature.com/scientificreports OPEN Deep learning on the 2‑dimensional Ising model to extract the crossover region with a variational autoencoder Nicholas Walker1*, Ka‑Ming Tam1,2 & Mark Jarrell1,2 The 2‑dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non‑vanishing feld case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising confgurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results on the confgurational magnetizations of the model. The performance of this method provides encouragement for the use of machine learning to extract meaningful structural information from complex physical systems where little a priori data is available. Machine learning (ML) and consequently data science as a whole have seen rapid development over the last decade or so, due largely to considerable advances in implementations and hardware that have made computa- tions more accessible. Conceptually, the ML approach can be regarded as a data modeling approach employing algorithms that eschew explicit instructions in favor of strategies based around pattern extraction and inference driven by statistical analysis. Tis presents a colossal opportunity for modern scientifc investigations, particu- larly numerical studies, as they naturally involve large data sets and complex systems where obvious explicit instructions for analysis can be elusive. Conventional approaches ofen neglect possible nuance in the structure of the data in favor of rather simple measurements that are ofen untenable for sufciently complex problems. Some ML methods such as inference methods have been routinely applied to certain physical problems, such as the maximum likelihood method and the maximum entropy method 1,2, but applications which utilizing ML methods have only recently attracted attention in the physical sciences, particularly for the study of interacting systems on both classical and quantum scales3. Tere is a unique opportunity to take advantage of the advances in ML algorithms and implementations to provide interesting new approaches to understanding physical data and even perhaps improve upon existing numerical methods 4. Outstanding problems involving the predictions of transition points and phase diagrams are also of great interest for treatment with ML methods. In order to utilize ML approaches for studying phase transitions, one must assume that there is some pat- tern change in the measured data across the phase transition. Fortunately, this is in fact exactly what happens in most phase transitions. Te widely adopted Lindemann parameter, for example, is essentially a measure of the deviations of atomic positions in the system from equilibrium positions and is ofen used to characterize the melting of a crystal structure 5. Similar form of pattern changes in the positions of the constituent atoms are ofen present in molecular systems in general. Perhaps more importantly, for some sufciently complex systems, their phase transitions do not have obvious order parameters, ofen prohibiting the detection of such pattern changes using conventional methods. Tis is not a hypothetical situation, indeed hidden orderings for some interesting materials, such as heavy fermion materials and cuprate superconductors, have been proposed for long time 6–8. Other systems may not even exhibit a true phase transition, but rather a crossover region where there is no singularity across diferent phases that can be difcult to characterize with conventional methods. A conventional phase transition can be identifed in two ways, with the frst being a singularity in a derivative of the free energy 1Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA. 2Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA. *email: nwalk19@ lsu.edu SCIENTIFIC REPORTS | (2020) 10:13047 | https://doi.org/10.1038/s41598-020-69848-5 1 Vol.:(0123456789) www.nature.com/scientificreports/ as proposed by Ehrenfest and the second being a broken symmetry exhibited by an order parameter as proposed by Landau. Unlike a conventional phase transition, a crossover is not identifed by a singularity in the free energy. Tere is also no broken symmetry in such a situation and thus no order parameter is associated with a crossover. Te order parameter and singularity in the free energy are presumably sharp and obvious features which can be rather easily identifed. Te absence of such features clearly present a challenging situation in the prediction of a crossover region by ML. ML is a new route of studying these systems by searching for hidden patterns in the measured data where readily applicable a priori information is in short supply. A viable ML method for detecting a crossover will fnd its use in many interesting systems related to the quantum phase transition9. While the quantum phase transition is a second order phase transition controlled by non-thermal parameters at zero temperature, all experiments and most numerical simulations are conducted at fnite albeit low temperatures for practical reasons. As a consequence of said thermal conditions, quantum critical points at low temperatures behave as crossover phenomenona. It is widely believed that many interesting materi- als, particularly high temperature cuprate superconductors, harbor a quantum critical point. An ML approach for detecting the crossover phenomenon can thus be an important tool for studying quantum critical points. Work has been done on various problems to characterize phase transitions in physical systems using ML methods, including the Ising model in the vanishing feld case 3,10–20. Tis work will use a similar approach to those seen in these papers, but will focus on the crossover regions that are introduced in the non-vanishing feld case of the 2-dimensional Ising model instead of seeking only the exactly known transition point in the vanishing feld case21. Tis is a somewhat more difcult problem, as there is no explicit transition to be found, but it remains an interesting problem nonetheless and possibly carries much greater implications for crossover regions in more complicated problems. Te Ising model itself is a mathematical model for ferromagnetism that is ofen explored in the feld of statistical mechanics in physics to describe magnetic phenomena22. Originally, the Ising model was developed to investigate magnetic phenomena, as mentioned earlier. With the discovery of electron spins, the model was designed to determine whether or not local interactions between magnetic spins could induce a large fraction of the electronic spins in a material to align in order to produce a macroscopic net magnetic moment. It is expressed in the form of a multidimensional array of spins si that represent a discrete arrangement of magnetic dipole 22 moments of atomic spins . Te spins are restricted to spin-up or spin-down alignments such that si 1, 1 . ∈ {− + } Te spins interact with their nearest neighbors with an interaction strength given by Jij for neighbors si and sj . Te spins can additionally interact with an applied external magnetic feld Hi (where the magnetic dipole moment µ has been absorbed). Te full Hamiltonian describing the system is thus expressed as H Jijsisj Hisi =− − (1) i,j i � � where i, j indicates a sum over adjacent spins. For Jij > 0 , the interaction between the spins is ferromagnetic, for Jij < 0 , the interaction between the spins is antiferromagnetic, and for Jij 0 , the spins are noninteracting. Fur- = thermore, if Hi > 0 , the spin at site i tends to prefer spin-up alignment, if Hi < 0 , the spin at site i tends to prefer spin-down alignment, and if Hi 0 , there is no external magnetic feld infuence on the spin at site i. Te model = has seen extensive use in investigating magnetic phenomena in condensed matter phhysics 24–29. Additionally, the model can be equivalently expressed in the form of the lattice gas model, described by the following Hamiltonian H 4J ninj µ ni =− − (2) i,j i � � where the external feld strength H is reinterpreted as the chemical potential µ , J retains its role as the interaction strength, and ni 0, 1 represents the lattice site occupancy. Te original Ising Hamiltonian can be recovered ∈{ } using the relation σi 2ni 1 up to a constant. Tis model describes a multidimensional array of lattice sites = − which can be either occupied or unoccupied by a hard shell atom, disallowing occupancy greater than one. Te frst term is then interpreted as a short-range attractive interaction term while the second is the fow of atoms between the system and the reservoir. Tis is a simple model of density fuctuation and the liquid-gas transfor- mations used primarily in chemistry, albeit ofen with modifcations 30,31. Additionally, modifed versions of the lattice gas models have been applied to binding behavior in biology 32–34. Typically, the model is studied in the case of Jij J 1 and the vanishing feld case Hi H 0 is of particu- = = = = lar interest for dimension d 2 since a phase transition is exhibited as the critical temperature is crossed. For ≥ two dimensions, the critical temperature can be identifed by exploiting Kramers-Wannier